Question

9.
(1 + tan²theta) cot theta
cosec theta
tan theta​

Answer 1

Proved

Step-by-step explanation:

Given

`((1+tan^2\theta). cot\theta)/(cosec^2 \theta)= tan \theta`

Required

Prove

In trigonometry:

`1 + tan^2\theta = sec^2\theta`

So, we have:

`(sec^2\theta * cot\theta)/(cosec^2 \theta)= tan \theta`

Express each identity as an inverse:

`((1)/(cos^2\theta) * (1)/(tan\theta))/((1)/(sin^2 \theta))= tan \theta`

Rewrite as:

`((1)/(cos^2\theta) * (1)/(tan\theta))/ (1)/(sin^2 \theta)= tan \theta`

Express tan as sin/cos:

`((1)/(cos^2\theta) * (1)/(sin\theta/cos\theta))/ (1)/(sin^2 \theta)= tan \theta`

`((1)/(cos^2\theta) * (1)/(sin\theta/cos\theta)) * sin^2 \theta= tan \theta`

`((1)/(cos^2\theta) * (cos\theta)/(sin\theta)) * sin^2 \theta= tan \theta`

`((1)/(cos\theta) * (1)/(sin\theta)) * sin^2 \theta= tan \theta`

`(sin^2 \theta)/(cos\theta*sin\theta)= tan \theta`

`(sin \theta)/(cos\theta)= tan \theta`

`tan \theta= tan \theta`

Proved